Section 2.9

Methods for Evaluating Indeterminate Forms

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Problem 1318

Find the value of the following:

lim⁡x→0sin⁡axsin⁡bx. \lim _{x \to 0}\frac{\sin ax}{\sin bx} .
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Problem 1319

Find the value of the following:

lim⁡x→0ch⁡x−cos⁡xx2. \lim _{x \to 0}\frac{\ch x-\cos x}{x^{2}} .
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Problem 1320

Find the value of the following:

lim⁡x→0tg⁡x−xx−sin⁡x. \lim _{x \to 0}\frac{\tg x-x}{x-\sin x} .
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Problem 1321

Find the value of the following:

lim⁡x→03tg⁡4x−12tg⁡x3sin⁡4x−12sin⁡x. \lim _{x \to 0}\frac{3\tg 4x-12\tg x}{3\sin 4x-12\sin x} .
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Problem 1322

Find the value of the following:

lim⁡x→π2tg⁡3xtg⁡x. \lim _{x \to \frac{\pi }{2}}\frac{\tg 3x}{\tg x} .
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Problem 1323

Find the value of the following:

lim⁡x→0xctg⁡x−1x2. \lim _{x \to 0}\frac{x\ctg x-1}{x^{2}} .
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Problem 1324

Find the value of the following:

lim⁡x→π4tg⁡x3−12sin⁡2x−1. \lim _{x \to \frac{\pi }{4}}\frac{\sqrt[3]{\tg x}-1}{2\sin ^{2}x-1} .
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Problem 1325

Find the value of the following:

lim⁡x→0x(ex+1)−2(ex−1)x3. \lim _{x \to 0}\frac{x(e^{x}+1)-2(e^{x}-1)}{x^{3}} .
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Problem 1326

Find the value of the following:

lim⁡x→01−cos⁡x2x2sin⁡x2. \lim _{x \to 0}\frac{1-\cos x^{2}}{x^{2}\sin x^{2}} .
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Problem 1327

Find the value of the following:

lim⁡x→0arcsin⁡2x−2arcsin⁡xx3. \lim _{x \to 0}\frac{\arcsin 2x-2\arcsin x}{x^{3}} .
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Problem 1328

Find the value of the following:

lim⁡x→01xx(aarctg⁡xa−barctg⁡xb). \lim _{x \to 0}\frac{1}{x\sqrt{x}}\left(\sqrt{a}\arctg \sqrt{\frac{x}{a}}-\sqrt{b}\arctg \sqrt{\frac{x}{b}}\right) .
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Problem 1329

Find the value of the following:

lim⁡x→0ax−asin⁡xx3. \lim _{x \to 0}\frac{a^{x}-a^{\sin x}}{x^{3}} .
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Problem 1330

Find the value of the following:

lim⁡x→1(xx−xln⁡x−x+1). \lim _{x \to 1}\left(\frac{x^{x}-x}{\ln x-x+1}\right) .
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Problem 1331

Find the value of the following:

lim⁡x→0ln⁡(sin⁡ax)ln⁡(sin⁡bx). \lim _{x \to 0}\frac{\ln (\sin ax)}{\ln (\sin bx)} .
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Problem 1332

Find the value of the following:

lim⁡x→0ln⁡(cos⁡ax)ln⁡(cos⁡bx). \lim _{x \to 0}\frac{\ln (\cos ax)}{\ln (\cos bx)} .
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Problem 1333

Find the value of the following:

lim⁡x→0cos⁡(sin⁡x)−cos⁡xx4. \lim _{x \to 0}\frac{\cos (\sin x)-\cos x}{x^{4}} .
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Problem 1334

Find the value of the following:

lim⁡x→01x(1th⁡x−1tg⁡x). \lim _{x \to 0}\frac{1}{x}\left(\frac{1}{\th x}-\frac{1}{\tg x}\right) .
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Problem 1335

Find the value of each of the following expressions:

lim⁡x→0\arsh(sh⁡x)−\arsh(sin⁡x)sh⁡x−sin⁡x, \lim _{x \to 0}\frac{\arsh (\sh x)-\arsh (\sin x)}{\sh x-\sin x} ,

where \arshx=ln⁡(x+1+x2)\arsh x=\ln (x+\sqrt{1+x^{2}}).

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Problem 1336

Find the value of each of the following expressions:

lim⁡x→+∞ln⁡xxε (ε<0). \lim _{x \to +\infty }\frac{\ln x}{x^{\varepsilon }} \ (\varepsilon <0) .
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Problem 1337

Find the value of each of the following expressions:

lim⁡x→+∞xneax (a>0,n>0). \lim _{x \to +\infty }\frac{x^{n}}{e^{ax}} \ (a>0, n>0) .
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Problem 1338

Find the value of each of the following expressions:

lim⁡x→0e−1x2x100. \lim _{x \to 0}\frac{e^{-\frac{1}{x^{2}}}}{x^{100}} .
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Problem 1339

Find the value of each of the following expressions:

lim⁡x→+∞x2e−0.01x. \lim _{x \to +\infty }x^{2}e^{-0.01x} .
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Problem 1340

Find the value of each of the following expressions:

lim⁡x→1−0ln⁡x⋅ln⁡(1−x). \lim _{x \to 1-0}\ln x \cdot \ln (1-x) .
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Problem 1341

Find the value of each of the following expressions:

lim⁡x→+0xεln⁡x (ε>0). \lim _{x \to +0}x^{\varepsilon }\ln x \ (\varepsilon >0) .
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Problem 1342

Find the value of each of the following expressions:

lim⁡x→+0xx. \lim _{x \to +0}x^{x} .
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Problem 1343

Find the value of each of the following expressions:

lim⁡x→+0xxx−1. \lim _{x \to +0}x^{x^{x}-1} .
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Problem 1344

Find the value of each of the following expressions:

lim⁡x→+0(xxx−1). \lim _{x \to +0}(x^{x^{x}}-1) .
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Problem 1345

Find the value of each of the following expressions:

lim⁡x→+0xk1+ln⁡x. \lim _{x \to +0}x^{\frac{k}{1+\ln x}} .
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Problem 1346

Find the value of each of the following expressions:

lim⁡x→1x11−x. \lim _{x \to 1}x^{\frac{1}{1-x}} .
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Problem 1347

Find the value of each of the following expressions:

lim⁡x→1(2−x)tg⁡πx2. \lim _{x \to 1}(2-x)^{\tg \frac{\pi x}{2}} .
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Problem 1348

Find the value of each of the following expressions:

lim⁡x→π4(tg⁡x)tg⁡2x. \lim _{x \to \frac{\pi }{4}}(\tg x)^{\tg 2x} .
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Problem 1349

Find the value of each of the following expressions:

lim⁡x→0(ctg⁡x)sin⁡x. \lim _{x \to 0}(\ctg x)^{\sin x} .
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Problem 1350

Find the value of each of the following expressions:

lim⁡x→+0(ln⁡1x)x. \lim _{x \to +0}\left(\ln \frac{1}{x}\right)^{x} .
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Problem 1351

Find the value of each of the following expressions:

lim⁡x→∞(tg⁡πx2x+1)1x. \lim _{x \to \infty }\left(\tg \frac{\pi x}{2x+1}\right)^{\frac{1}{x}} .
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Problem 1352

Find the value of each of the following expressions:

lim⁡x→a(tg⁡xtg⁡a)ctg⁡(x−a). \lim _{x \to a}\left(\frac{\tg x}{\tg a}\right)^{\ctg (x-a)} .
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Problem 1353

Find the value of each of the following expressions:

lim⁡x→0(ax−xln⁡abx−xln⁡b)1x2. \lim _{x \to 0}\left(\frac{a^{x}-x\ln a}{b^{x}-x\ln b}\right)^{\frac{1}{x^{2}}} .
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Problem 1354

Find the value of each of the following expressions:

lim⁡x→0(1x−1ex−1). \lim _{x \to 0}\left(\frac{1}{x}-\frac{1}{e^{x}-1}\right) .
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Problem 1355

Find the value of each of the following expressions:

lim⁡x→1(1ln⁡x−1x−1). \lim _{x \to 1}\left(\frac{1}{\ln x}-\frac{1}{x-1}\right) .
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Problem 1356

Find the value of each of the following expressions:

lim⁡x→0(ctg⁡x−1x). \lim _{x \to 0}\left(\ctg x-\frac{1}{x}\right) .
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Problem 1357

Find the value of each of the following expressions:

lim⁡x→0(1ln⁡(x+1+x2)−1ln⁡(1+x)). \lim _{x \to 0}\left(\frac{1}{\ln (x+\sqrt{1+x^{2}})}-\frac{1}{\ln (1+x)}\right) .
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Problem 1358

Find the value of each of the following expressions:

lim⁡x→aax−xax−a (a>0). \lim _{x \to a}\frac{a^{x}-x^{a}}{x-a} \ (a>0) .
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Problem 1359

Find the value of each of the following expressions:

lim⁡x→0(1+x)1x−ex. \lim _{x \to 0}\frac{(1+x)^{\frac{1}{x}}-e}{x} .
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Problem 1360

Find the value of each of the following expressions:

lim⁡x→0(a+x)x−axx2 (a>0). \lim _{x \to 0}\frac{(a+x)^{x}-a^{x}}{x^{2}} \ (a>0) .
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Problem 1361

Find the value of each of the following expressions:

lim⁡x→+∞(2πarctg⁡x)x. \lim _{x \to +\infty }\left(\frac{2}{\pi }\arctg x\right)^{x} .
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Problem 1362

Find the value of each of the following expressions:

lim⁡x→+∞(th⁡x)x. \lim _{x \to +\infty }(\th x)^{x} .
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Problem 1363

Find the value of each of the following expressions:

lim⁡x→0(arcsin⁡xx)1x2. \lim _{x \to 0}\left(\frac{\arcsin x}{x}\right)^{\frac{1}{x^{2}}} .
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Problem 1363.1
[NO TASK]
Problem 1363.2
[NO TASK]
Problem 1363.3
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Problem 1363.4
[NO TASK]
Problem 1364

Find the value of each of the following expressions:

lim⁡x→0((1+x)1xe)1x. \lim _{x \to 0}\left(\frac{(1+x)^{\frac{1}{x}}}{e}\right)^{\frac{1}{x}} .
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Problem 1365

Find the value of each of the following expressions:

lim⁡x→0(2πarccos⁡x)1x. \lim _{x \to 0}\left(\frac{2}{\pi }\arccos x\right)^{\frac{1}{x}} .
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Problem 1366

Find the value of each of the following expressions:

lim⁡x→0(cos⁡xch⁡x)1x2. \lim _{x \to 0}\left(\frac{\cos x}{\ch x}\right)^{\frac{1}{x^{2}}} .
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Problem 1367

Find the value of each of the following expressions:

lim⁡x→0ln⁡ch⁡xch⁡xm−ch⁡xn. \lim _{x \to 0}\frac{\ln \ch x}{\sqrt[m]{\ch x}-\sqrt[n]{\ch x}} .
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Problem 1368

Find the value of each of the following expressions:

lim⁡x→0(1+ex2)ctg⁡x. \lim _{x \to 0}\left(\frac{1+e^{x}}{2}\right)^{\ctg x} .
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Problem 1368.1
[NO TASK]
Problem 1369

Find the value of each of the following expressions:

lim⁡x→+∞(x3+x2+x+13−x2+x+1⋅ln⁡(ex+x)x). \lim _{x \to +\infty }\left(\sqrt[3]{x^{3}+x^{2}+x+1}-\sqrt{x^{2}+x+1} \cdot \frac{\ln (e^{x}+x)}{x}\right) .
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Problem 1370

Find the value of each of the following expressions:

lim⁡x→+∞[(x+a)1+1x−x1+1x+a]. \lim _{x \to +\infty }\left[(x+a)^{1+\frac{1}{x}}-x^{1+\frac{1}{x+a}}\right] .
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Problem 1371

If, as x→0x \to 0, the curve y=f(x)y=f(x) passes through the origin (0,0) [lim⁡x→0f(x)=f(0)=0](0,0) \ \left[\lim_{x \to 0}f(x)=f(0)=0\right], and has inclination angle α\alpha there, find

lim⁡x→0yx. \lim _{x \to 0}\frac{y}{x} .
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Problem 1372

If, as x→+0x \to +0, the curve y=f(x)y=f(x) passes through the origin (0,0) [lim⁡x→+0f(x)=0](0,0) \ \left[\lim_{x \to +0}f(x)=0\right], and when 0<x<ε0<x<\varepsilon, this curve lies entirely within the acute angle formed by the two lines y=−kxy=-kx and y=kx (k≠∞)y=kx \ (k \neq \infty ), prove that

lim⁡x→+0xf(x)=1. \lim _{x \to +0}x^{f(x)}=1 .
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Problem 1373

Prove: if the second-order derivative f′′(x)f''(x) of the function f(x)f(x) exists, then

f′′(x)=lim⁡h→0f(x+h)+f(x−h)−2f(x)h2. f''(x)=\lim _{h \to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^{2}} .
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Problem 1373.1
[NO TASK]
Problem 1373.2
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Problem 1374

Investigate the possibility of applying L'Hospital's rule to the following examples:

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(a)

lim⁡x→0x2sin⁡1xsin⁡x\displaystyle \lim_{x \to 0}\frac{x^{2}\sin \dfrac {1}{x}}{\sin x};

(b)

lim⁡x→∞x−sin⁡xx+sin⁡x\displaystyle \lim_{x \to \infty }\frac{x-\sin x}{x+\sin x};

(c)

lim⁡x→+∞e−2x(cos⁡x+2sin⁡x)+e−x2sin⁡2xe−x(cos⁡x+sin⁡x)\displaystyle \lim_{x \to +\infty }\frac{e^{-2x}(\cos x+2\sin x)+e^{-x^{2}}\sin^{2}x}{e^{-x}(\cos x+\sin x)};

(d)

lim⁡x→+∞1+x+sin⁡xcos⁡x(x+sin⁡xcos⁡x)esin⁡x\displaystyle \lim_{x \to +\infty }\frac{1+x+\sin x\cos x}{(x+\sin x\cos x)e^{\sin x}}.

Problem 1375

Given a circular segment with chord bb and height (sagitta) hh, and an isosceles triangle inscribed in this segment; if RR is held fixed while the arc of the segment tends to zero, find the ratio of the area of the segment to the area of the inscribed isosceles triangle. Use the result obtained to derive the approximate formula for the area of the segment:

S≈23bh. S \approx \frac{2}{3}bh .
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